Finite Group Databasebeta

← #28644

Subgroups — #28644

Order 256.

370 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#488C211yesmaximal, characteristic
2C2×#446C211yesmaximal, characteristic, complemented
3#2592C211yesmaximal, characteristic
4#2584C211yesmaximal, characteristic, complemented
5#2066C212yesmaximal, complemented
6#2065C212yesmaximal
7#1989C211yesmaximal, characteristic, complemented
8#1983C211yesmaximal, characteristic
9#1936C212yesmaximal
10#1935C212yesmaximal, complemented
11C4×#106C211yesmaximal, characteristic, complemented
12C2×#125C2211yescharacteristic
13C2×#124C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15#488—24no—
16#446—24no—
17C2×#106C2212yes—
18C2×C4⋊C8C2212yes—
19C2×C4⋊C8C2211yescharacteristic
20C2×#103C2211yescharacteristic
21C2×#103C2212yes—
22C2×#103C2211yescharacteristic
23C2×#102C2211yescharacteristic
24C2×#102C2212yes—
25C2×#102C2211yescharacteristic
26C2×#97C2212yes—
27C2×C4×C8C2211yescharacteristic
28#398—21no—
29#397—21no—
30#395—21no—
31#393—21no—
32#391C2212yes—
33#390C2212yes—
34#389—22no—
35#389C2212yes—
36#389C2212yes—
37#388—22no—
38C4×C4⋊C4C2211yescharacteristic
39C4×C4⋊C4C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43C22×Q8D411yescharacteristic
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C22×C8C2312yes—
47C22×C8C2311yescharacteristic
48C22×C8C2311yescharacteristic
49#125—22no—
50#125—22no—
51#124—22no—
52#124—22no—
53C4×Q8—22no—
54C4×Q8—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4D412yes—
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C42—21no—
70C2×C42—21no—
71C2×C42C2312yes—
72C2×C42C2311yescharacteristic
73#106—24no—
74C4⋊C8—24no—
75C4⋊C8—22no—
76#103—22no—
77#103—22no—
78#103—24no—
79#102—22no—
80#102—22no—
81#102—24no—
82#97—24no—
83C4×C8D414yes—
84#95—21no—
85#95—21no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95—22no—
90C24—21no—
91C2×Q8—22no—
92C2×Q8—21no—
93C2×Q8—22no—
94C2×Q8—21no—
95C2×Q8—22no—
96C2×Q8—42no—
97C2×D4—21no—
98C2×D4—22no—
99C2×D4—21no—
100C2×D4—22no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4C2×D412yes—
107C22×C4C2×D412yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2411yescharacteristic
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8C2×D412yes—
117C2×C8C2×D412yes—
118C4⋊C4—22no—
119C4⋊C4—24no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—22no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—44no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—82no—
131C4⋊C4#4114yes—
132C22⋊C4—44no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C22⋊C4—82no—
136C42—22no—
137C42—22no—
138C42—81no—
139C42—81no—
140C42C2×D412yes—
141C42#4112yes—
142C23—41no—
143C23—42no—
144C23—42no—
145C23—41no—
146C23—41no—
147C23C22×D411yescharacteristic
148Q8—44no—
149Q8—41no—
150Q8—41no—
151Q8—42no—
152D4—41no—
153D4—41no—
154D4—42no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4#12112yes—
178C2×C4#12112yes—
179C2×C4C22×D411yescharacteristic
180C2×C4C2×#4112yes—
181C2×C4C2×#4111yescharacteristic
182C2×C42−1+412yes—
183C8—22no—
184C8—22no—
185C8—42no—
186C22—82no—
187C22—82no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22C2×#12111yescharacteristic
192C22#54812yes—
193C22#57512yes—
194C22#57711yescharacteristic
195C22#57711yescharacteristic
196C4—22no—
197C4—22no—
198C4—44no—
199C4—42no—
200C4—42no—
201C4—82no—
202C4—81no—
203C4—81no—
204C4—81no—
205C4#53211yescharacteristic
206C4#53611yescharacteristic
207C4#55612yes—
208C2—161no—
209C2#254511yescharacteristic
210C2#305411yescharacteristic
211C2#306011yescharacteristic
212C2#337112yes—
213C2#337612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#446C2semidirect
4. #2584C2semidirect
5. #2066C2semidirect
7. #1989C2semidirect
10. #1935C2semidirect
11. C4×#106C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#4882
C2×#4462
#25922
#25842
#20662
#20652
#19892
#19832
#19362
#19352
C4×#1062